Dynamics of a delayed reaction-diffusion predator-prey model with the effect of the toxins

Nomenclature

1.   Introduction

Electric hydraulic servo system (EHSS) is widely used in industry applications due to its high power ratio and fast response ^{[1,2,3,4,5,6]}. With the rapid development of technology, more severe control requirements are put forward for EHSS, and high-performance control becomes eagerly needed to address its behavioral nonlinearity and uncertainties, such as flow nonlinearity, pressure dynamic nonlinearity, uncertain parameters, unmolded nonlinearity, and external disturbances ^[7,8,9,10].

Nowadays, many researchers focus on advanced control strategies such as robust adaptive control ^[11,12], sliding mode control ^[13,14], back-stepping control ^[15,16], robust integral of the sign of the error (RISE) ^[17,18] for EHSS. Yue and Yao ^[17] proposed an adaptive robust integral of the sign of the error control (ARISE), which can adjust the robust gain online through adaptive method to solve the potential high gain feedback of symbolic function. Yao et al. ^[19] designed a novel ARSE to address noise pollution in the acquisition of acceleration signal, which can compensate the error of friction model and other bounded disturbances ^[20]. The above literature shows that RISE/ARISE control greatly improves the trajectory tracking accuracy of EHSS. However, those controllers regard the parameter adaptive error, unmolded error, and external disturbance as lumped disturbance, which severely limits the control accuracy. Control strategy based on disturbance observer can compensate the influence of disturbance and uncurtains effectively, which has been used in the field of control theory and engineering ^{[21,22,23,24,25,26]}. Especially, the extended state observer (ESO), which is the core of ADRC and has been widely used in disturbance estimation and suppression ^{[27,28,29,30,31,32]}.

In this paper, the sign function is replaced with a modified arctangent function to smooth the nonlinearity of sign function. Then, the state and external disturbance can be estimated by ESO, respectively. Finally, the residual observation error is compensated to further enhance the tracking accuracy by ARISE. The Lyapunov theory proves the EHSS can achieve asymptotic s. Simulation results show the proposed controller has a better performance in maxi-mum tracking error, average tracking error and standard deviation of the tracking error.

2.   Dynamic model of electro-hydraulic servo system

The typical working principle of EHSS is shown in Figure 1. The double rod symmetrical hydraulic cylinder is controlled by servo valve to drive the load.

In this paper, the force balance equation of EHSS can be given as

where m_p and x_p represent mass and displacement of load respectively; P_L is the pressure difference; A_p is the effective area of the piston; k is the spring stiffness; f_v is the combined coefficient; $f\left(x_{\mathrm{p}}, \dot{x}_{\mathrm{p}}, t\right)$ indicates the lumped uncertain; Pressure dynamics of the two chambers are given by ^[15].

where

The pressure difference dynamics between the two chambers is expressed as follows

where β_e1, β_e2 are the effective elastic modulus in two chamber and β_e1 = β_e2 = β_e; V₁ = V₀₁ + A_p; V₂ = V₀₂ - A_p·x_p represents the control volume of return chamber; V₀₁ and V₀₂ are the initial volumes of the two chambers respectively; C_t is the internal leakage coefficient of the cylinder; Q₁ (t) and Q₂ (t) are the oil flowrate of the two chamber of the cylinder respectively; q₁(t) and q₂(t) are model errors because of internal leakage in the two chambers; g₁ is the pressure difference at the oil inlet and g₂ is the pressure difference at the oil outlet; P_s is the supplied pressure; P_r is the return pressure, k_t is the total gain of the servo valve; u is the control voltage and s(u) is expressed as

To make it smooth and differentiable, the sign function s(u) is replaced by Eq (7).

Thus, g₁, g₂ in Eq (4) can be rewritten as Eq (8).

Substituting Eqs (2)-(5) into Eq (1), and thus

Rewritten Eq (9), and thus,

where $f^{\prime}$ is the approximation error caused by using the continuously differentiable function farctan(u).

According to Eq (1)

Substituting Eq (11) into Eq (10), thus

Define state variables as $\boldsymbol{x} = {\left[{{x_1}, {x_2}, {x_3}} \right]^{\rm T}} = {\left[{{x_{\text{p}}}, {{\dot x}_{\text{p}}}, {{\ddot x}_{\text{p}}}} \right]^{\rm T}}$ and output variables as $y = {x_1} = {x_{\text{p}}}$, so the state space model of EHSS can be expressed as

where $\theta_1 = A_{\mathrm{p}} \beta_e k_t\left(\frac{g_1^{\prime}}{V_1}+\frac{g_1^{\prime}}{V_2}\right) / m_{\mathrm{p}}, \theta_2 = \beta_e C_{\mathrm{t}}\left(\frac{1}{V_1}+\frac{1}{V_2}\right) k / m_{\mathrm{p}}$, $\theta_3 = \left(A_{\mathrm{p}}^2 \beta_e\left(\frac{1}{V_1}+\frac{1}{V_2}\right)+f_{\mathrm{v}} \beta_e C_{\mathrm{t}}\left(\frac{1}{V_1}+\frac{1}{V_2}\right)+k\right) / m_{\mathrm{p}}$, $\theta_4 = \left(m_{\mathrm{p}} \beta_e C_{\mathrm{t}}\left(\frac{1}{V_1}+\frac{1}{V_2}\right)+f_{\mathrm{v}}\right) / m_{\mathrm{p}}, \Delta = \left(\beta_e C_{\mathrm{t}}\left(\frac{1}{V_1}+\frac{1}{V_2}\right) f+\dot{f}+f^{\prime}+A_{\mathrm{p}} \beta_e\left(\frac{q_1}{V_1}+\frac{q_1}{V_2}\right)\right) / m_{\mathrm{p}}$.

In practice, the parameters m, k, β_e and C_t may not be known accurately, so it is necessary to consider the uncertainties of these parameters. Define vector as $\boldsymbol{\theta } = \left[{{\theta _1}, {\theta _2}, {\theta _3}, {\theta _4}} \right]$ and improve the tracking performance of the system through the adaptive method.

The purpose of the system controller is to design a bounded control input u so that y = x₁ can track the desired trajectory y_d(t) = x_1d(t). Therefore, the following assumptions should be given.

Assumption 1: The desired trajectory x_1d is five times differentiable and each is bounded. In practice, the load pressure of hydraulic cylinder meets 0 < P_L < P_s.

Assumption 2: The range of parametric uncertainties is

where $\theta_{\min } = \left[\theta_{1 \min }, \cdots, \theta_{4 \min }\right]^{T}$, $\theta_{\max } = \left[\theta_{1 \max }, \cdots, \theta_{4 \max }\right]^{T}$ are known.

Assumption 3: The time-varying perturbation Δ (t) of Eq (10) is smooth enough so that

where δ₁, δ₂ are known positive constants.

3.   Extend state observer design

ESO can estimate the uncertainty disturbance comprehensively, so we use ESO to estimate the lumped disturbance and compensate it feed forward to achieve better tracking accuracy.

Define x₄ = Δ(t), ${\dot x_4} = \delta \left(t \right)$ and the expanded state space can be written as

Defining $\hat{x} = x-\hat{x}$ as the estimation error of x, where $\hat{x}$ represents the estimated value of x.

According to the expanded state space model, the ESO is designed as

where ω₀ is the bandwidth of the extended state observer, $\hat{\theta}_{i}$ is the estimated value of the unknown parameters θ_i, I = 1, 2, 3, 4.

The dynamic equation of observation error can be obtained by subtracting Eq (17) an Eq (18):

where $\boldsymbol{\tilde{\theta}} = \left[\tilde{\theta}_{1}, \tilde{\theta}_{2}, \tilde{\theta}_{3}, \tilde{\theta}_{4}\right]^{\mathrm{T}}$, $\boldsymbol{ \varPhi}_{1} = \left[u, x_{1}, x_{2}, x_{\xi}, x_{4}\right]^{T}$.

Let

Then Eq (17) can be written as

Since matrix A is a Hurwitz matrix, and there is a positive definite symmetric matrix P which satisfies the following equation

The symmetric positive definite matrix P is:

4.   Projection mapping and parameter adaptation

Considering Eq (14) and Eq (15), the discontinuous projection can be defined as ^[11].

where $\hat{\alpha}$ denote the estimate of $\alpha$ and $\tilde{\alpha}$ denote the estimate error, $\tilde{\alpha} = \hat{\alpha}-\alpha, i = 1, 2, 3, 4$.

Using the adaptation law as follow:

where Γ is the diagonal positive definite adaptation rate matrix, τ is an adaptation function. For any adaption function τ, the adaptation Eq (25) satisfies follow ^[3]:

5.   Design of adaptive robust integral of the sign of the error controller

Defining the following error variables

where x_1d is the given trajectory; k₁ k₂ k₃ are the positive feedback gain and r is the auxiliary error signal. Because r contains the differentiation of acceleration, it is considered to be unmeasurable in practice and only used for auxiliary design. According to Eq (28), r has the following expansion:

Dividing Eq (29) by θ₁, and thus

where ${\alpha _1} = {1 \mathord{\left/ {\vphantom {1 {{\theta _1}}}} \right. } {{\theta _1}}}$, ${\alpha _2} = {{{\theta _2}} \mathord{\left/ {\vphantom {{{\theta _2}} {{\theta _1}}}} \right. } {{\theta _1}}}$, ${\alpha _3} = {{{\theta _3}} \mathord{\left/ {\vphantom {{{\theta _3}} {{\theta _1}}}} \right. } {{\theta _1}}}$, ${\alpha _4} = {{{\theta _4}} \mathord{\left/ {\vphantom {{{\theta _4}} {{\theta _1}}}} \right. } {{\theta _1}}}$.

The model-based controller is designed as follows:

where ${\hat \alpha = }{\left[{{{\hat \alpha }_1}, {{\hat \alpha }_2}, {{\hat \alpha }_3}, {{\hat \alpha }_4}} \right]^{\rm T}}$ represents the estimated value of ${\alpha } = {\left[{{\alpha _1}, {\alpha _2}, {\alpha _3}, {\alpha _4}} \right]^{\rm T}}$; ${k_r} > 0$ is the gain of controller; $\beta > 0$ is the robust gain; ${u_a}$ is the feedforward model compensation, u_s1 is the linear robust feedback term, u_s2 is the RISE control term, ${{\boldsymbol{\varPhi }}_2} = {\left[{{{\dddot x}_{1{\text{d}}}}, {x_{1{\text{d}}}}, {{\dot x}_{1{\text{d}}}}, {{\ddot x}_{1{\text{d}}}}} \right]^{\rm T}}$.

Substituting Eq (31) into Eq (30) and note that $d = - {\hat \alpha _1}\hat \Delta + {\alpha _1}\Delta$

${\tilde \alpha = \hat \alpha } - {\alpha }$, Substitute Eq (25) into Eq (32), thus

The overall structure of the designed control strategy is shown in Figure 2.

Lemma 1: Define variable L(t) as

Define auxiliary function as

According to ^[8], if the gain β satisfies the following inequality, then the auxiliary function P(t) is always positive definite.

Theorem 1: Using the adaptive law Eq (25), and adaptive function $\tau = - r{{\boldsymbol{\dot \Phi }}_2}$, and the robust gain β satisfies inequality Eq (36) as well as the feedback gains k₁, k₂, k₃, k_r are sufficient to ensure that the matrix ${\Lambda }$ defined below is positive definite, the adaptive robust integral of the sign of the error controller Eq (31) can make all signals bounded in the closed-loop system, and the system obtains asymptotic output tracking, i.e., ${z_1} \to 0$ as $t \to \infty$.

where ${k_4} = \max \left({{{{\dot {\boldsymbol{\varPhi}} }}_2}^{\rm T}\boldsymbol{\varGamma }{{\boldsymbol{\varPhi} }_2}} \right) + {k_3} + {k_r} - C$, ${k_5} = A{k_1}$, ${k_6} = - \left({A - B{k_2}} \right)$, ${k_7} = - \left({B + C{k_3}} \right)$, $\max \left(\cdot \right)$ represents the maximum value of the matrix.

Proof: see Appendix A.

6.   Simulation validation

The nominal value of the physical parameters of the valve controlled symmetrical hydraulic cylinder are shows in Table 1. The following controllers are compared by simulation to validate the effectiveness of the designed controller.

(1) Controller I: ESO based ARISE This is the controller designed in this paper. The controller parameters are selected as: k₁ = 200, k₂ = 180, k₃ = 0.08, k_r = 0.012, β = 0.05, ω = 100. According to the nominal value of the parameters, the estimated boundary of unknown parameter α are given as: α_min = [5 × 10^-5 0.03 2.5 8 × 10^-3] and α_max = [7 × 10^-5 0.07 3.5 20.2 × 10^-3]. The initial estimates of α is set as $\hat \alpha (0)$ = [6.5×10^-5 0.04 3 8.2×10^-3] and Γ is set as diag[1 × 10^-15 2 × 10^-2 0.001 5 × 10^-10].

(2) Controller II: ARISE without ESO. Compared to the controller II, there is no ESO compensation term and the other parameters are same to controller I. That is only u_α in Eq (18) is replaced as: ${u_a} = {\hat \alpha _1}{\dddot x_{1{\text{d}}}} + {\hat \alpha _2}{x_{1{\text{d}}}} + {\hat \alpha _3}{\dot x_{1{\text{d}}}} + {\hat \alpha _4}{\ddot x_{1{\text{d}}}} = {\hat \alpha ^{\rm T}}{{\boldsymbol{\varPhi }}_2}$

(3) Controller III: PI controller. The parameters are set as k_P = 410 and k_I = 10, which are the optimal solutions after repeated debugging.

(4) Controller IV: BP neutral network PID controller. The structure of the neutral network is 3-5-3, and the learning rate $\eta = 0.28$, inertia coefficient α = 0.3.

The desired trajectories are designed as three cases: normal motion with the motion trajectory

${x_{\text{d}}}\left(t \right) = {{10\arctan \left[{\sin \left({\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} \mathord{\left/ {\vphantom {{10\arctan \left[{\sin \left({\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} {0.7854}}} \right. } {0.7854}}$ mm, fast level motion with the motion trajectory ${x_{\text{d}}}\left(t \right) = {{10\arctan \left[{\sin \left({4\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} \mathord{\left/ {\vphantom {{10\arctan \left[{\sin \left({4\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} {0.7854}}} \right. } {0.7854}}$ mm and low-level motion with the motion trajectory ${x_{\text{d}}}\left(t \right) = {{10\arctan \left[{\sin \left({0.2\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} \mathord{\left/ {\vphantom {{10\arctan \left[{\sin \left({0.2\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} {0.7854}}} \right. } {0.7854}}$ mm. The external disturbance is designed as $f\left(t \right) = {{20\arctan \left[{\sin \left({0.8\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} \mathord{\left/ {\vphantom {{20\arctan \left[{\sin \left({0.8\pi t} \right)} \right]\left({1 - {e^{ - t}}} \right)} {0.7854}}} \right. } {0.7854}}$ N.

To compare the tracking responses of each controller quantitatively, three performance indices including maximum absolute value of the tracking error M_e, average tracking error ${\mu _{\text{e}}}$, standard deviation of the tracking error σ_e, which were defined in are adopted to evaluate ^[20].

(1) Case I-normal level motion

The four controllers are tested for a normal motion trajectory x_d(t) = 10arctan[sin(πt)] [1-exp(-t)]/0.7854 mm. The tracking performance are shown in Figures 4-6, the performance indices of the four controllers is shown in Table 2. From the Figures 4-6 and Table 1, it is obviously that the valve controlled symmetrical cylinder has the best tracking performance under the controller designed in this paper than other controllers. From Table 2, the amplitudes of steady-state tracking error of the controller III and controller IV are both about 0.6 mm, while controller I is about 0.003 mm and controller II is about 0.01 mm, it shows that the ARISE can deal with nonlinear and uncertainties and disturbance well but PI controller just has some robustness. By comparing the performance indices in Table 2 and tracking error in Figure 6 of controller I and controller II, it can be seen that controller I is better than controller II in all indices obviously, which indicates that the parameter adaptation in Figure 6 and ESO compensates for both parametric and uncertain lumped disturbance are effective. The control input u of controller I showed in Figure 3 is continuous and smooth, which makes it easy to implement in practice.

(2) Case II-low level motion

In this case, a low level reference trajectory x_d(t) = 10arctan[sin(0.2πt)][1-exp(-t)]/0.7854 mm is tested. The results are shown in Figures 7-9 and the performance indices are listed in Table 3. From Figure 7 and Table 3, the amplitudes of steady-state tracking error of the controller III and controller IV are both about 0.076mm, while controller I and controller II both are about 9.5 × 10^-5 mm, which shows that the ARISE can also deal with nonlinear and uncertainties and disturbance well in low level reference trajectory. By comparing the performance indices of controller I and controller II in Table 3, it can be seen that the maximum absolute value of the tracking error M_e of the two controllers almost the same, but the average tracking error μ_e, and standard deviation of the tracking error σ_e of controller I are 6.536 × 10^-6 mm and 8.833 × 10^-6 mm respectively which are better than that of 2.87371 × 10^-5 and 1.02822 × 10^-5 of controller II obviously, which further validates the effectiveness of the desired parameter adaptation and ESO compensation. The parameter adaptation of controller I are omitted. The control input u of controller I showed in Figure 8 is continuous and there is slight high-frequency vibration. So the controller designed in this paper has the best tracking performance on low level reference trajectory than other controllers too.

(3) Case III-fast level motion

A faster level reference trajectory x_d(t) = 10arctan[sin(4πt)][1-exp(-t)]/0.7854 mm is tested in this case. The results are shown in Figured 10-12 and the performance indices are listed in Table 4. From Figure 12 and Table 4, the amplitudes of steady-state tracking error of the controller III and controller IV are both about 1.24 mm, while controller I 1.34 × 10^-2 mm and controller II 4.20256 × 10^-2 mm, which shows that the ARISE can better deal with nonlinear and uncertainties and disturbance in fast level reference trajectory than PI controller too. In addition, comparing the performance indices of controller I and controller II in Table 4, it can be seen that the maximum absolute value of the average tracking error μ_e of the controller I is 4.14 × 10^-3 mm and the controller II is 1.63032 × 10^-2 mm, the standard deviation of the tracking error σ_e of controller I are 5.53 ×10^-3 mm and the controller II is 1.30355 × 10^-2 mm. These further validate the effectiveness of the desired parameter adaptation and ESO compensation. The control input u of controller I showed in Figure 11 is continuous and smooth. The result verifies that the controller designed in this paper still has high tracking accuracy in tracking performance on fast level reference trajectory.

7.   Conclusions

In this paper, an ARISE with ESO controller is proposed for EHSS to address parametric uncertainties, uncertainty nonlinearities and unmolded disturbances. The proposed ARISE can compensate the dynamics uncertainties, thus guaranteeing asymptotic tracking and improving the adaptability and safety of EHSS. ESO can effectively estimate the state and lumped uncurtains. Simulation results shows that ARISE with ESO can obtain high tracking accuracy and better performance in tracking desired trajectory under all working conditions.

Conflict of interest

The authors declare there is no conflict of interest.

Appendix A

Define a Lyapunov function

It is Obvious that V is positive definite. The derivative of V is:

Substituting Eqs (25) and (28), L(t)and P(t), into Eq (A.2).

where $\boldsymbol{\eta} = \left[z_1, z_2, z_3, r\right]^{\mathrm{T}}, \lambda \min (\boldsymbol{\varLambda})$ is the minimum eigenvalue of matrix Λ, therefore $V \in L_{\infty}$ and $W \in L_2$, so z₁, z₂, z₃ and r are bounded. According to the assumptions 1 and 2, all states of the system are bounded so the actual control input u is bounded. According to Eqs (28) and (33), the derivative of W is bounded, so W is uniformly continuous. According to Barbarat's lemma, W→0 as t→∞, so the conclusion of the theorem 1 can be deduced, theorem 1 is proofed.